MAKRO EKONOMI

ECON 402: Solow’s Growth Model 1

Advanced Macroeconomics, ECON 402

Solow’s Growth Model

1 Why are we interested in Economic Growth?

Tracking the differentials between average real incomes, standards of living across economies

reveal a vast chasm between “haves” and “have nots” economies. The stark difference is

magnified when we consider that the World Economy has enlarged over the past decades.

The issue is more than a redistribution of wealth from one set of nations to another, since

what we as Economist wish to do is to provide a mechanism through which indepen-

dent economies can become self-sufficient, and become productive members within the

Economic System at large. Further, at the microeconomic level, these differences are as-

sociated with nutritional and educational differences, variations in infant mortality rates

and life-expectancies, all measures of wellbeing that much of Wealthy Nations has grown

accustomed to as a norm and right, which makes the stark differences between wealthy

and poor economies unacceptable. It is these welfare concerns that draw us from focusing

on counteracting short term fluctuations, but instead focus on our understanding how

long run growth can be achieved.

You have obtained a glimpse into considerations of growth in Ramsey (1928). We

will now consider the model by Solow (1956). It is the standard bearer in growth models

through which most other growth models are compared against. The premise of Solow’s

(1956) model is the observe differences in output per capita growth both inter-temporally

and across economies cannot be explained by capital accumulation, where capital accu-

mulation operates through the usual channel of enhancing production. The shortfall of

the model is in the fact that it ignores the positive externalities of capital, and capital

accumulation, and either completely ignores, or relegates as exogenous, other sources that

can explain growth differentials. Nonetheless, it is these shortfalls that permit us to build

upon his original findings.

TUGAS ANALISIS OPTIMASI UAS


2 Solow’s Growth Model

2.1 Assumptions

The four variables of the model are output (Y ), capital (K), labour (L), and effectiveness

of labour (A).

1. The production function takes the general form,

Y (t) = F (K(t), A(t)L(t)) (1)

where t denotes time. Further, notice that

(a) The production function is not dependent on time. What does this say?

(b) A and L are multiplicatively separable, and we say it is Labour-Augmenting, or

effects labour only directly. This manner of specifying the production function

implies that at equilibrium, K/Y is a constant. This is not far fetched insofar

as this ratio never reveals any clear trend. But this assumption does say how

“effectiveness” affect labour productivity.

2. The Production function exhibits Constant Returns to Scale. In other words, for

any constant c ≥ 0,

F (cK, cAL) = cF (K,AL) (2)

The implication of this assumption are as follows:

(a) The assumption of constant returns to scale says essentially that the economy

has exhausted the gains to specialization, which is a reasonable assumption

if we are concerned with a developed economy. However, within the context

of a economy in the midst of the transition, an increase in the components of

production may still yield gains to specialization, which in turn yield increasing

returns to scale.

(b) Notice also that because the production function focuses on Labour and Capi-

tal, and neglects resources such as land, and natural resource, the assumptions

on the production function may not allow for decreasing returns to scale. De-

pending on the focus of the economy, for example one that relies on natural

resources and land intensive industries, increasing inputs may lead to decreas-

ing returns to scale. Nonetheless, there are historically relevant examples where



the lack of natural resource and land had not been presented a hindrance to

the growth or development process.

(c) Another term to describe the constant returns to scale is that it assumes that

the production function is a linearly homogeneous function. Let c = 1/AL,

then we can write the production function as,

F (cK, cAL) = F

(K

AL, 1

)=

1

ALF (K,AL)

⇒ F (K,AL)

AL=

Y

AL

where Y/AL is output per unit of effective labour, and K/AL is capital per

unit of labour. If we define k = K/AL, y = Y/AL and f(k) = F (k, 1), then

the production function can be written as,

y = f(k) (3)

which is the production function define in terms of per unit of effective labour,

as a function of capital per unit of effective labour. This has the interpretation

that output is dependent on the capital per unit of effective labour, and not

on the size of the economy.

3. The production function, f(k) satisfies the following,

f(0) = 0

f ′(k) > 0

f ′′(k) < 0

which is says that the marginal product of capital is increasing and concave in

capital K. In addition, the function must satisfy the Inada condition,

limk→0

f ′(k) = ∞

limk→∞

f ′(k) = 0

which implies that at small levels of capital per unit of effective labour, the marginal

product of capital is very large, and becomes very small when the quantity of capital

becomes very large. This additional assumption essentially ensures the production

function is well behaved, by ensuring the path that the economy takes in the dynamic



sense will not diverge. An example of such a production function is one that is of

the Cobb-Douglas form,

F (K,AL) = Kα(AL)1−α

where α ∈ (0, 1).

4. The initial values of the variables K, L and A are given, and the latter two variables,

labour and knowledge grows at a constant rate,

∂L(t)

∂t= L(t) = nL(t) (4)

∂A(t)

∂t= A(t) = gA(t) (5)

which implies that both variables grow at an exponential rate. This is because at an

exponential rate of growth, for example with an exponential rate of growth labour in

any period is L(t) = L(0)ent, where as noted L(0) is known. Then L = L(0)entn =

nL(t).

5. Output produced is either consumed or saved (and invested as in the Ramsey model)

at a rate of s, which in turn is exogenously given and constant across time. Further,

it is assumed that a unit of investment from savings yields a unit of capital. Finally,

capital depreciates at a rate δ. Thus the relationship of capital with respect to the

other inputs can be defined as,

K = sY (t)− δK(t) (6)

Note that this does not describe a growth rate such as that for Labour and Effec-

tiveness.

Provide a critique of the model beyond the points noted in your text, regarding

the “lack of realism” of this model.

2.2 Solving the Model

Since the two of the inputs growth rates in the economy is given exogenously, all that is

left for us to determine is to understand the evolution of the capital, or the capital per

unit of effective labour. Notice there is no optimization involved in this economy.



Given that both Labour and Effectiveness is growing, it is likely that Capital would

likewise be the same. Consequently, we will focus on the growth rate of Capital per unit

of effective labour, k. Then from equations (4), (5), (6) and the fact that k = K/AL,

k(t) =K(t)

A(t)L(t)

⇒ k(t) =K(t)

A(t)L(t)− K(t)(A(t)L(t) + A(t)L(t))

(A(t)L(t))2

⇒ k(t) =K(t)

A(t)L(t)− K(t)

A(t)L(t)

A(t)

A(t)− K(t)

A(t)L(t)

L(t)

L(t)(7)

⇒ k(t) =sY (t)− δK(t)

A(t)L(t)− (n+ g)k(t)

= sy(t)− δk(t)− (n+ g)k(t)

⇒ k(t) = sf(k(t))− (n+ g + δ)k(t) (8)

What equation (8) says is that the evolution of capital per unit effective labour is invest-

ment rate net of the rate required to keep up (break-even) with the growth rate of the

other inputs. The intuition behind the rate is that given capital is constantly depreciat-

ing, an economy would need at the minimum to replace this capital lost just to maintain

the same level of output. This lead in turn to the second point, that the previous state-

ment assumed that there were no growth in the effectiveness of labour, and labour itself.

Insofar as both inputs are themselves growing, capital must grow to maintain the effec-

tiveness of labour, and absorb (which provides a critique of the model. Does it consider

unemployment?) the additional workforce. The diagrammatic representation of this is

below in figure 1. Letting k∗ denote the point where the rate of gross investment is equal

to the rate of maintenance, the implications of equation (8) is that if k < k∗, capital per

unit of effective labour will continue to grow towards k∗, while if k > k∗, capital per unit

of effective labour will be less than replacement, and capital per unit of effective labour

will fall towards k∗. In other words, at all other initial points of k 6= k∗, the economy

will converge to k∗. It is only at k = k∗ that the economy is at stead state equilibrium.

However, note that should the economy begin at k(0) = 0, it will not elevate towards k∗

since k = k(0) is also a trivial steady state equilibrium. Can you see why?



Figure 1: Relationship between Components of k

k

k-

6

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��

k∗

(n+ g + δ)k

sf(k)

When k converges to k∗, sy(t) = (n+ g + δ)k(t),

sy(t) = (n+ g + δ)k(t)

sY (t)− δK(t)

A(t)L(t)= (n+ g)

K(t)

A(t)L(t)

K(t)

K(t)= (n+ g)

You can likewise show that the growth rate of effective labour is n + g (Show yourself

that it is true.). Since we have a constant rate of return, this would mean that the



growth rate of output is likewise n+ g. To see this,

Y (t) = F (K(t), A(t)L(t))

= F (K(0)e(n+g)t, A(0)L(0)e(n+g)t)

= e(n+g)tF (K(0), A(0)L(0))

= e(n+g)tY (0)

⇒ Y (t) = e(n+g)tY (0)(n+ g) = Y (t)(n+ g)

⇒ Y (t)

Y (t)= n+ g

Finally, show yourself that the growth rate of output and capital per unit of

labour is g.

On the aggregate, what the Solow model highlights is that regardless of any starting

point (with the exception k = 0), an economy will tend towards a “balanced growth path”

where the components of a economy grow at a constant rate. It is important to note that

the economy does not stagnate, as noted in the Ramsey model, but grows at a constant

rate, and this growth rate is dependent on the rate of technological progress augmented

on the labour force.

2.3 Some Comparative Statics

Given that the growth rates are exogenously determined, the variable that is within the

“jurisdiction” of the social planner is the saving rate s, and it is interesting how its

change could affect the equilibrium growth rate of capital, as well as the equilibrium level

of capital. It is worth your while to consider how a government could affect changes in

societal savings rate.

From figure 1, and equation (8), it is clear that should the government or social

planner be able to raise savings rate (to a rate of s′ > s), it only affects the investment

component only. From equation (8), it is clear that investment growth would outweigh

the maintenance of capital to keep pace with the effectiveness of labour, thereby leading

to a growth in capital. This increase growth in capital leads to a increased growth in

output produced per unit of labour. To see this, note first that Y/L = (Y/AL)(AL/L) =

Ay = Af(k). We know that at the Balanced Growth Path, Y/L grows at a rate of g. In

the case of an increase in investments due to an increase an increase in savings rate, then



the growth rate of Y/L = Af(k) must exceed g, since k is growing because K is growing.

However, using the same arguments of before, the economy would eventually tend towards

the new balanced growth path. Intuitively, as capital increases due to the new savings

rate, an economy has the capability to produce more output, however, once this new

capabilities have been exhausted, there cannot be a higher level of steady state growth of

output beyond the given technology that is available. Eventually, the growth in capital

is purely as replacement for the new practice. This highlights that an increase in savings,

and consequently an increase in investment and an increase in the level of output and

output per unit of labour, but not a longer growth rate. In short, increase in savings

leads to a level effect, not a growth effect. This analysis is reflected in the figure below.

Note that the only manner in which growth effect can be achieved is to technological

advancements that affect the effectiveness of labour, in other words, the growth rate of

A.

Figure 2: Effect of a Change in Savings Rate on k

k

k-

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��

k∗

(n+ g + δ)k

sf(k)

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......................................................................s′f(k)

k′∗

We can extend the examination of changes in savings rate on consumption of the

constituents of the economy. Strictly speaking, the Solow model does not contend with



the individual consumer, nor their interaction with the firms, in other words it is not a

general equilibrium model, we can augment the treatment by realizing that what is not

saved and invested into future production capabilities is consumed. Therefore, the rate

of consumption is 1− s, so that if savings rate changes discretely, so would consumption

rate. However, since the output eventually rises with capital accumulation, it would lead

to an increased level of consumption from the initial dip. However, we cannot be certain

that this increase is greater than before the exogenous change in savings rate. To see this,

note first that from equation (8), and focusing focusing on the original balanced growth

path value of capital k∗,

c∗ = f(k∗(s, n, g, δ))− (n+ g + δ)k∗(s, n, g, δ)

⇒ ∂c∗

∂s= (fk − (n+ g + δ))

∂k∗

∂s

so that consumption at the balanced growth path increases or otherwise depends on

whether the bracketed item is positive, negative or equal to zero. In the event where it is

equal to zero implies that the economy is at a level of capital k∗ on its balanced growth

path so that changes in savings rates will not affect its consumption level. This level of

capital stock is known as the Golden Rule level of capital stock.

2.4 Long Run Effects & the Speed of Convergence

To understand the long run effects of a change in savings rate on output, first note that,

y∗ = f(k∗(s, n, g, δ))

⇒ ∂y∗

∂s= f ′(k∗)

∂k∗(s, n, g, δ)

∂s

Next note that since in the steady state, k = 0, which as defined by equation (8), we have

sf(k∗(s, n, g, δ)) = (n+ g + δ)k∗(s, n, g, δ)

f(k∗) + sf ′(k∗)∂k∗

∂s= (n+ g + δ)

∂k∗

∂s∂k∗

∂s=

f(k∗)

n+ g + δ − sf ′(k∗)

Therefore,

∂y∗

∂s=

f(k∗)f ′(k∗)

n+ g + δ − sf ′(k∗)(9)



It is not completely clear from equation (9) that as savings rate increases, it would induce

an eventual increase in output, since it is dependent on whether the net growth rate of

effective labour is greater or less than sf ′(k∗). In fact if the denominator is equal to zero,

the relationship between output per unit of effective labour, y∗ and saving rate is defined

at all! The relationship however is clearer if we write it in terms of elasticities, which can

be achieved by multiplying both sides of the equation by s/y∗ = s/f(k∗),

∂y∗

∂s

s

y∗=

f(k∗)f ′(k∗)

n+ g + δ − sf ′(k∗)s

f(k∗)

=f ′(k∗)(n+ g + δ)k∗

(n+ g + δ)f(k∗)− f ′(k∗)(n+ g + δ)k∗

=f ′(k∗)

f(k∗)/k∗ − f ′(k∗)

Noting that the elasticity of output per unit of effective labour with respect to capital is

εk = ∂f(k∗)∂k∗

k∗

f(k∗)= f ′(k∗)k∗/f(k∗),

∂y∗

∂s

s

y∗=

f ′(k∗)k∗/f(k∗)

1− f ′(k∗)k∗/f(k∗)(10)

=εk

1− εk(11)

Assuming the economy is competitive with no externalities as mentioned in the setup of

the model, each unit of capital would be valued at its marginal product, which in turn

means that the total value of capital is k∗f(k∗). This then allows us to interpret the the

elasticity of output with respect to each unit of capital per unit of effective labour to be

the share of total value of output that is accruing to capital per unit of effective labour.

This formula can then be used to examine the effectiveness of changes in savings rate on

output growth of an economy (Read your text.). Using this model, what has been found

is that savings has at best a moderate effect, principally due to the fact that the measured

elasticity of most economies are low. This in turn implies that the marginal change of

sf ′k is large, which implies in turn that the sf(k) is very concave, so that for a change

in capital per unit of effective labour, the consequent change in equilibrium steady state

output is small.

However, the preceding analysis does not examine the speed at which the new steady

state is attained. In other words, even if the output increase were large, the gains from

that increase might take a disproportionately long time to attain. To examine the speed



with which an economy could converge towards the new steady state equilibrium, from

equation (8), note that k ≡ k(k). Put another way, the path of capital per unit of effective

labour is dependent on the level of capital per unit of labour itself. Then by a first order

Taylor Series Expansion of the relationship we have about k = k∗,

k ≈ k(k∗) +∂k(k)

∂k

∣∣∣∣∣k=k∗

(k − k∗)

=∂k(k)

∂k

∣∣∣∣∣k=k∗

(k − k∗)

= (sf ′(k∗) + (n+ g + δ))(k − k∗)

=

((n+ g + δ)k∗f ′(k∗)

f(k∗)− (n+ g + δ)

)(k − k∗)

⇒ k ≈ −(1− εk)(n+ g + δ)(k − k∗) (12)

which says that the rate at which k converges towards k∗ is proportional to the distance

between the location of k from k∗, and that proportion is −(1 − εk)(n + g + δ). To see

that, first note that ˙(k(t)− k∗) = ∂(k(t)−k∗)∂t

, since k∗ is just a constant. Therefore,

k(t)− k∗ = e−(1−εk)(n+g+δ)t(k(0)− k∗)∂(k(t)− k∗)

∂t= −((1− εk)(n+ g + δ)t)e−(1−εk)(n+g+δ)t(k(0)− k∗)

⇒˙(k(t)− k∗)

(k(t)− k∗)= −(1− εk)(n+ g + δ) (13)

Then typical empirical estimates of the parameters, the researcher would be able to ex-

amine the convergence rates, assuming that the model is an adequate representation of

the economy in question (Read your text for some examples).

Read your text on the implications of the Solow model on the question of

growth, and the empirical findings.

References

Ramsey, F. P. (1928): “A Mathematical Theory of Saving,” The Economic Journal,

pp. 543–559.

Solow, R. (1956): “A Contribution to the Theory of Economic Growth,” Quarterly

Journal of Economics, 70, 65–94.


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